| L(s) = 1 | − 2·2-s + 3·4-s − 2·7-s − 4·8-s + 9-s + 2·11-s + 4·14-s + 5·16-s − 2·18-s − 4·22-s − 6·23-s − 8·25-s − 6·28-s − 2·29-s − 6·32-s + 3·36-s + 18·37-s − 10·43-s + 6·44-s + 12·46-s − 3·49-s + 16·50-s − 4·53-s + 8·56-s + 4·58-s − 2·63-s + 7·64-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.755·7-s − 1.41·8-s + 1/3·9-s + 0.603·11-s + 1.06·14-s + 5/4·16-s − 0.471·18-s − 0.852·22-s − 1.25·23-s − 8/5·25-s − 1.13·28-s − 0.371·29-s − 1.06·32-s + 1/2·36-s + 2.95·37-s − 1.52·43-s + 0.904·44-s + 1.76·46-s − 3/7·49-s + 2.26·50-s − 0.549·53-s + 1.06·56-s + 0.525·58-s − 0.251·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.988976394201284184542048138074, −8.187970805601434868644449073054, −7.943842402331196870218149621155, −7.66306734096358645414521611121, −6.92523802114892024920431126793, −6.41126060400557503794066668857, −6.21971640217724100178398085245, −5.64616578159812240665779595163, −4.82031804393880400172708827249, −3.91125550867391516486622446076, −3.68892919486575087815854496997, −2.69270944971101382127846043632, −2.09070899691935900455323863207, −1.24538182000997094239238289843, 0,
1.24538182000997094239238289843, 2.09070899691935900455323863207, 2.69270944971101382127846043632, 3.68892919486575087815854496997, 3.91125550867391516486622446076, 4.82031804393880400172708827249, 5.64616578159812240665779595163, 6.21971640217724100178398085245, 6.41126060400557503794066668857, 6.92523802114892024920431126793, 7.66306734096358645414521611121, 7.943842402331196870218149621155, 8.187970805601434868644449073054, 8.988976394201284184542048138074
